Dominated Operators
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Dominated Operators
Author: A.G. Kusraev
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-03-09
The notion of a dominated or rnajorized operator rests on a simple idea that goes as far back as the Cauchy method of majorants. Loosely speaking, the idea can be expressed as follows. If an operator (equation) under study is dominated by another operator (equation), called a dominant or majorant, then the properties of the latter have a substantial influence on the properties of the former . Thus, operators or equations that have "nice" dominants must possess "nice" properties. In other words, an operator with a somehow qualified dominant must be qualified itself. Mathematical tools, putting the idea of domination into a natural and complete form, were suggested by L. V. Kantorovich in 1935-36. He introduced the funda mental notion of a vector space normed by elements of a vector lattice and that of a linear operator between such spaces which is dominated by a positive linear or monotone sublinear operator. He also applied these notions to solving functional equations. In the succeedingyears many authors studied various particular cases of lattice normed spaces and different classes of dominated operators. However, research was performed within and in the spirit of the theory of vector and normed lattices. So, it is not an exaggeration to say that dominated operators, as independent objects of investigation, were beyond the reach of specialists for half a century. As a consequence, the most important structural properties and some interesting applications of dominated operators have become available since recently.
Limit Operators and Their Applications in Operator Theory
This text has two goals. It describes a topic: band and band-dominated operators and their Fredholm theory, and it introduces a method to study this topic: limit operators. Band-dominated operators. Let H = [2(Z) be the Hilbert space of all squared summable functions x : Z -+ Xi provided with the norm 2 2 X IIxl1 :=L I iI . iEZ It is often convenient to think of the elements x of [2(Z) as two-sided infinite sequences (Xi)iEZ. The standard basis of [2(Z) is the family of sequences (ei)iEZ where ei = (. . . ,0,0, 1,0,0, . . . ) with the 1 standing at the ith place. Every bounded linear operator A on H can be described by a two-sided infinite matrix (aij)i,jEZ with respect to this basis, where aij = (Aej, ei)' The band operators on H are just the operators with a matrix representation of finite band-width, i. e. , the operators for which aij = 0 whenever Ii - jl > k for some k. Operators which are in the norm closure ofthe algebra of all band operators are called band-dominated. Needless to say that band and band dominated operators appear in numerous branches of mathematics. Archetypal examples come from discretizations of partial differential operators. It is easy to check that every band operator can be uniquely written as a finite sum L dkVk where the d are multiplication operators (i. e.
Vector Lattices and Intergal Operators
Author: Semën Samsonovich Kutateladze
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-12-06
The theory of vector lattices, stemming from the mid-thirties, is now at the stage where its main achievements are being summarized. The sweeping changes of the last two decades have changed its image completely. The range of its application was expanded and enriched so as to embrace diverse branches of the theory of functions, geometry of Banach spaces, operator theory, convex analysis, etc. Furthermore, the theory of vector lattices was impregnated with principally new tools and techniques from other sections of mathematics. These circumstances gave rise to a series of mono graphs treating separate aspects of the theory and oriented to specialists. At the same time, the necessity of a book intended for a wider readership, reflecting the modern diretions of research became clear. The present book is meant to be an attempt at implementing this task. Although oriented to readers making their first acquaintance with vector-lattice theory, it is composed so that the main topics dealt with in the book reach the current level of research in the field, which is of interest and import for specialists. The monograph was conceived so as to be divisible into two parts that can be read independently of one another. The first part is mainly Chapter 1, devoted to the so-called Boolean-valued analysis of vector lattices. The term designates the applica tion of the theory of Boolean-valued models by D. Scott, R. Solovay and P.